Multi-Criteria Decision Analysis www.it-ebooks.info Multi-Criteria Decision Analysis Methods and Software Alessio Ishizaka Reader in Decision Analysis, Portsmouth Business School University of Portsmouth, UK Philippe Nemery Senior Research Scientist, SAP Labs – China, Shanghai, PRC www.it-ebooks.info This edition first published 2013 C 2013 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data Ishizaka, Alessio Multi-criteria decision analysis : methods and software / Alessio Ishizaka, Philippe Nemery pages cm Includes bibliographical references and index ISBN 978-1-119-97407-9 (cloth) Multiple criteria decision making Multiple criteria decision making–Data processing Decision support systems I Nemery, Philippe II Title T57.95.I84 2013 003 56–dc23 2013004490 A catalogue record for this book is available from the British Library ISBN: 978-1-119-97407-9 Typeset in 10/12pt Times by Aptara Inc., New Delhi, India 2013 www.it-ebooks.info Contents Foreword xi Acknowledgements xiii General introduction 1.1 Introduction 1.2 Decision problems 1.3 MCDA methods 1.4 MCDA software 1.5 Selection of MCDA methods 1.6 Outline of the book References Part I FULL AGGREGATION APPROACH Analytic hierarchy process 2.1 Introduction 2.2 Essential concepts of AHP 2.2.1 Problem structuring 2.2.2 Priority calculation 2.2.3 Consistency check 2.2.4 Sensitivity analysis 2.3 AHP software: MakeItRational 2.3.1 Problem structuring 2.3.2 Preferences and priority calculation 2.3.3 Consistency check 2.3.4 Results 2.3.5 Sensitivity analysis 2.4 In the black box of AHP 2.4.1 Problem structuring 2.4.2 Judgement scales 2.4.3 Consistency 2.4.4 Priorities derivation 2.4.5 Aggregation www.it-ebooks.info 1 5 11 13 13 13 14 16 18 19 20 20 21 22 24 25 27 27 28 31 33 39 vi CONTENTS 2.5 Extensions of AHP 2.5.1 Analytic hierarchy process ordering 2.5.2 Group analytic hierarchy process 2.5.3 Clusters and pivots for a large number of alternatives 2.5.4 AHPSort References Analytic network process 3.1 Introduction 3.2 Essential concepts of ANP 3.2.1 Inner dependency in the criteria cluster 3.2.2 Inner dependency in the alternative cluster 3.2.3 Outer dependency 3.2.4 Influence matrix 3.3 ANP software: Super Decisions 3.3.1 Problem structuring 3.3.2 Assessment of pairwise comparison 3.3.3 Results 3.3.4 Sensitivity analysis 3.4 In the black box of ANP 3.4.1 Markov chain 3.4.2 Supermatrix References Multi-attribute utility theory 4.1 Introduction 4.2 Essential concepts of MAUT 4.2.1 The additive model 4.3 RightChoice 4.3.1 Data input and utility functions 4.3.2 Results 4.3.3 Sensitivity analysis 4.3.4 Group decision and multi-scenario analysis 4.4 In the black box of MAUT 4.5 Extensions of the MAUT method 4.5.1 The UTA method 4.5.2 UTAGMS 4.5.3 GRIP References MACBETH 5.1 Introduction 5.2 Essential concepts of MACBETH 5.2.1 Problem structuring: Value tree www.it-ebooks.info 40 41 44 48 50 54 59 59 59 60 63 64 67 68 69 70 73 74 76 76 78 80 81 81 81 83 89 89 93 94 95 97 98 98 105 111 112 114 114 114 115 CONTENTS 5.2.2 Score calculation 5.2.3 Incompatibility check 5.3 Software description: M-MACBETH 5.3.1 Problem structuring: Value tree 5.3.2 Evaluations and scores 5.3.3 Incompatibility check 5.3.4 Results 5.3.5 Sensitivity analysis 5.3.6 Robustness analysis 5.4 In the black box of MACBETH 5.4.1 LP-MACBETH 5.4.2 Discussion References Part II OUTRANKING APPROACH vii 117 118 122 122 122 125 127 127 127 131 131 133 133 135 PROMETHEE 6.1 Introduction 6.2 Essential concepts of the PROMETHEE method 6.2.1 Unicriterion preference degrees 6.2.2 Unicriterion positive, negative and net flows 6.2.3 Global flows 6.2.4 The Gaia plane 6.2.5 Sensitivity analysis 6.3 The Smart Picker Pro software 6.3.1 Data entry 6.3.2 Entering preference parameters 6.3.3 Weights 6.3.4 PROMETHEE II ranking 6.3.5 Gaia plane 6.3.6 Sensitivity analysis 6.4 In the black box of PROMETHEE 6.4.1 Unicriterion preference degrees 6.4.2 Global preference degree 6.4.3 Global flows 6.4.4 PROMETHEE I and PROMETHEE II ranking 6.4.5 The Gaia plane 6.4.6 Influence of pairwise comparisons 6.5 Extensions of PROMETHEE 6.5.1 PROMETHEE GDSS 6.5.2 FlowSort: A sorting or supervised classification method References 137 137 137 138 142 143 146 148 149 149 151 153 155 157 158 160 162 163 164 166 167 168 170 170 172 177 ELECTRE 7.1 Introduction 180 180 www.it-ebooks.info viii CONTENTS 7.2 Essentials of the ELECTRE methods 7.2.1 ELECTRE III 7.3 The Electre III-IV software 7.3.1 Data entry 7.3.2 Entering preference parameters 7.3.3 Results 7.4 In the black box of ELECTRE III 7.4.1 Outranking relations 7.4.2 Partial concordance degree 7.4.3 Global concordance degree 7.4.4 Partial discordance degree 7.4.5 Outranking degree 7.4.6 Partial ranking: Exploitation of the outranking relations 7.4.7 Some properties 7.5 ELECTRE-Tri 7.5.1 Introduction 7.5.2 Preference relations 7.5.3 Assignment rules 7.5.4 Properties References 180 183 189 190 191 193 194 194 195 196 196 197 199 203 204 204 205 207 207 210 GOAL, ASPIRATION OR REFERENCE-LEVEL APPROACH 213 Part III TOPSIS 8.1 Introduction 8.2 Essentials of TOPSIS References 215 215 215 221 Goal programming 9.1 Introduction 9.2 Essential concepts of goal programming 9.3 Software description 9.3.1 Microsoft Excel Solver 9.4 Extensions of the goal programming 9.4.1 Weighted goal programming 9.4.2 Lexicographic goal programming 9.4.3 Chebyshev goal programming References 222 222 222 227 227 228 228 230 232 234 10 235 Data Envelopment Analysis Jean-Marc Huguenin 10.1 10.2 Introduction Essential concepts of DEA 10.2.1 An efficiency measurement method www.it-ebooks.info 235 236 236 CONTENTS 10.2.2 A DEA case study 10.2.3 Multiple outputs and inputs 10.2.4 Types of efficiency 10.2.5 Managerial implications 10.3 The DEA software 10.3.1 Building a spreadsheet in Win4DEAP 10.3.2 Running a DEA model 10.3.3 Interpreting results 10.4 In the black box of DEA 10.4.1 Constant returns to scale 10.4.2 Variable returns to scale 10.5 Extensions of DEA 10.5.1 Adjusting for the environment 10.5.2 Preferences 10.5.3 Sensitivity analysis 10.5.4 Time series data References Part IV 11 INTEGRATED SYSTEMS Multi-method platforms 11.1 Introduction 11.2 Decision Deck 11.3 DECERNS 11.3.1 The GIS module 11.3.2 The MCDA module 11.3.3 The GDSS module 11.3.4 Integration References Appendix: Linear optimization A.1 Problem modelling A.2 Graphical solution A.3 Solution with Microsoft Excel Index ix 237 247 248 249 252 254 255 257 262 263 266 268 268 268 269 270 270 275 277 277 278 278 279 281 284 286 287 288 288 289 289 293 www.it-ebooks.info Foreword The growing recognition that decision makers will often try to achieve multiple, and usually conflicting, objectives has led during the last three decades to the development of multi-criteria decision analysis (MCDA) This is now a vast field of research, with its scientific community and its specialized journals, as well as a large and growing number of real-world applications, for supporting both public policy making and decisions by private corporations Students and practitioners coming to the field, however, will be surprised by the plethora of alternative methods, overloaded by the array of software available, and puzzled by the diversity of approaches that an analyst needs to choose from For precisely these reasons, this book is a very welcome event for the field Alessio Ishizaka and Philippe Nemery have managed to provide an accessible, but rigorous, introduction to the main existing MCDA methods available in the literature There are several features of the book that are particularly innovative First, it provides a balanced assessment of each method, and positions them in terms of the type of evaluation that the decision requires (a single choice among alternatives, the ranking of all alternatives, the sorting of alternatives into categories, or the description of consequences) and the level of preference information that each method requires (from utility functions to no preference information) This taxonomy helps both researchers and practitioners in locating adequate methods for the problems they need to analyze Second, the methods are presented with the right level of formulation and axiomatization for an introductory course This makes the book accessible to anyone with a basic quantitative background Readers who wish to learn in greater depth about a particular method can enjoy the more advanced content covered ‘in the black box’ of each chapter Third, the book illustrates each method with widely available and free software This has two major benefits Readers can easily see how the method works in practice via an example, consolidating the knowledge and the theoretical content They can also reflect on how the method could be used in practice, to facilitate real-world decision-making processes Fourth, instructors using the book, as well as readers, can benefit from the companion website (www.wiley.com/go/multi criteria decision analysis) and the availability of software files and answers to exercises www.it-ebooks.info xii FOREWORD This book should therefore be useful reading for anyone who wants to learn more about MCDA, or for those MCDA researchers who want to learn more about other MCDA methods and how to use specialized software to support multi-criteria decision making Gilberto Montibeller Department of Management London School of Economics www.it-ebooks.info MULTI-METHOD PLATFORMS 11.3.2 281 The MCDA module The MCDA module aims to structure a decision problem (i.e define the criteria and alternatives), evaluate the alternatives and weight the criteria, thereby solving the problem First, the decision maker has to choose from the following decision aid methods: r ranking methods: ◦ MAUT (see Chapter 4) ◦ MAVT (multi-attribute value theory) ◦ AHP (see Chapter 2) ◦ PROMETHEE (see Chapter 6) ◦ TOPSIS (see Chapter 8) r sorting method: ◦ FlowSort (see Chapter 6.5.2) It is not necessary for the input data to be precisely defined Probabilistic distributions and ‘fuzzy’ numbers can be used in the case of uncertainty Different graphical and tabular tools are implemented to introduce probabilistic input/output data (i.e density distributions) The weights and the evaluations can be roughly defined by means of distributions such as the normal, uniform, log-normal and delta distributions This is illustrated in Figure 11.2 Different graphical and tabular tools are also Figure 11.2 The probability menu in DECERNS Reproduced by permission of Boris Yatsalo www.it-ebooks.info 282 MULTI-CRITERIA DECISION ANALYSIS Figure 11.3 Performance table Reproduced by permission of Boris Yatsalo available to present fuzzy input/output data Fuzzy numbers, for example, triangular, trapezoidal, piecewise linear or singleton, can be used to represent the evaluations and weights The following fuzzy and probabilistic multi-criteria methods have been implemented in DECERNS (Yatsalo et al 2011a): r PROMAA (Probabilistic Multi-criteria Acceptability Analysis); r FMAA (Fuzzy Multi-criteria Acceptability Analysis); r Fuzzy MAVT (Fuzzy Multi Attribute Value Utility) In these methods, the user can enter the data via a performance table (Figure 11.3) or a value tree (Figure 11.4) Changing the value of the weight or preference functions performs a sensitivity analysis As an illustrative example, refer to the Case Study 4.1, where the choice of five smartphones is evaluated based on four criteria In this section, we have chosen, by way of illustration, to regroup the screen size and storage size criteria into the ‘technical parameters’subgroup Figure 11.4 Determination of the value function Reproduced by permission of Boris Yatsalo www.it-ebooks.info MULTI-METHOD PLATFORMS 283 Figure 11.5 Tree structure of a decision problem in DECERNS, showing the Direct weighting dialogue as well as the performances of SP5 Reproduced by permission of Boris Yatsalo As illustrated in Figure 11.5, the tree structure gives a user-friendly representation of the decision problem The task (choice of a smartphone), criteria (price, customer reviews, etc.) and alternatives (SP1, SP2, etc.) can be viewed in this tree The dialogue boxes have been added to allow the introduction of direct weights and performance of alternatives There is another view of the decision problem, which displays the performance table as in Figure 11.3 Figure 11.5 shows that the weight determination can be achieved in various ways The user can choose from: r direct weight determination; r the SWING method; r ranking of the weights; r rating of the weights; r pairwise comparison of the weights as with the AHP method (see Chapter 2) The preference value functions of the criteria can be ‘drawn’ easily according to the decision maker’s preference (see Figure 11.4) The user has a choice between piecewise linear functions and exponential functions The final results are illustrated in Figure 11.6 www.it-ebooks.info 284 MULTI-CRITERIA DECISION ANALYSIS Figure 11.6 Results according to the MAUT method Reproduced by permission of Boris Yatsalo The user can perform two different types of sensitivity analysis by: r modifying the weights of the criteria or displaying the ‘Line weights’ representation which illustrates which weight value in a specifically chosen criterion changes the ranking (Figure 11.7); r modifying the shape of the utility function of a criterion and analysing the corresponding change in the ranking (Figure 11.8) The user can easily change the decision aid method, for example, when changing the decision aid in the TOPSIS method, the user needs to redefine the model-specific parameters, such as the preference functions in PROMETHEE Figure 11.9 represents the results obtained with the TOPSIS method while defining identical weight values (the user needs to consider the meaning of the parameters for each method as they are often different) Figure 11.10 shows the sensitivity analysis of the result obtained with the TOPSIS method when changing the weight values This differs significantly from the analysis obtained with the MAUT method 11.3.3 The GDSS module The DECERNS project has a specific GDSS module which permits the creation and administration of online surveys The results of these surveys are automatically collected and analysed This feature is essential when alternatives have to be assessed by various decision makers www.it-ebooks.info MULTI-METHOD PLATFORMS 285 Figure 11.7 Criterion analysis window Reproduced by permission of Boris Yatsalo Figure 11.8 Value function analysis window Reproduced by permission of Boris Yatsalo www.it-ebooks.info 286 MULTI-CRITERIA DECISION ANALYSIS Figure 11.9 Representation of the results obtained with the TOPSIS method Reproduced by permission of Boris Yatsalo 11.3.4 Integration As mentioned in the introduction, all modules are fully integrated This implies that the user can define areas in a map and define them as alternatives of the decision problem The alternatives from a map can automatically be transferred into the performance table and vice versa This integration is a strong advantage of the decision support system Figure 11.10 The criterion analysis window for the TOPSIS results www.it-ebooks.info MULTI-METHOD PLATFORMS 287 References Bisdorff, R., Meyer, P., and Roubens, M (2008) Rubis: A bipolar-valued outranking method for the choice problem 4OR, 6(2), 143–165 Dias, L., and Climaco, J (2000) Additive aggregation with variable interdependent parameters: the VIP analysis software Journal of the Operational Research Society, 51, 1070–1082 Dias, L., and Mousseau, V (2003) IRIS: A DSS for multiple criteria sorting problems Journal of Multi-Criteria Decision Analysis, 12(4–5), 285–298 Figueira, J., Greco, S., and Slowinski, R (2009) Building a set of additive value functions representing a reference preorder and intensities of preference: GRIP method European Journal of Operational Research, 195(2), 460–486 Grabisch, M., Meyer, P., and Kojadinovic, I (2012) Kappalab R-Script Retrieved June 2012, from Kappalab: http://cran.r-project.org/web/packages/kappalab/kappalab.pdf Grebenkov, A., Yatsalo, B., Sullivan, T., and Linkov, I (2007) DECERNS: Methodology and Software for Risk-Based Land Use Planning and Decision Support (in EnviroInfo) Aachen: Shaker Verlag Greco, S., Mousseau, V., and Slowinski, R (2008) Ordinal regression revisted: Multiple criteria ranking using a set of additive value functions European Journal of Operational Research, 191, 416–436 Gritsyuk, S., Yatsalo, B., Babutski, A., Mirzeabasov, O., and Didenko, V (2011) Multicriteria decision analysis with the use of DECERNS DSS MCDM Conference 2011 Ishizaka, A., Nemery, P., and Lidouh, K (2013) Location selection for the construction of a casino in the greater London region: A triple multi-criteria approach Tourism Management, 34(1), 211–220 Sullivan, T., Yatsalo, B., Grebenkov, A., and Linkov, I (2009) Decision Evaluation for Complex Risk Network Systems (Decerns) software tool In A Marcomini, G W Suter II, and A Critto (eds), Decision Support Systems for Risk-Based Management of Contaminated Sites New York: Springer Science + Business Media Veneziano, T., Bigaret, S., and Meyer, P (2009) Diviz: An MCDA components workflow execution engine Euro XXIII: 23td Euroepan Conference on Operational Research Bonn, Germany Yatsalo, B (2007) Decision support system for risk based land management and rehabilitation of radioactively contaminated territories: PRANA approach International Journal of Emergency Management, 4(3), 504–523 Yatsalo, B., Didenko, V., Tkachuk, A., and Gritsyuk, S (2010a) Application to land use planning International Journal of Information Systems and Social Change, 1(1), 11–30 Yatsalo, B., Gritsyuk, S., Didenko, V., Vasilevskaya, M., Mirzeabasov, O., and Babutski, A (2010b) Land-use planning and risk management with the use of web-based multi-criteria spatial decision support system (DECERNS) 25th Mini Euro-Conference: Uncertainty and Robustness in Planning and Decision Making Coimbra, Portugal Yatsalo, B., Gritsyuk, S., Mirzeabasov, O., and Vasilvskaya, M (2011a) Uncertainty treatment within multicriteria decision analysis with the use of acceptability concept Control of Big Systems, 32, 5–30 Yatsalo, B., Sullivan, T., Didenko, V., and Linkov, I (2011b) Environmental risk management for radiological accidents: Integrating risk assessment and decision analysis for remediation at different spatial scales Integrated Environmental Assessment and Management, 7(3), 393–395 www.it-ebooks.info Appendix Linear optimization A.1 Problem modelling Linear optimization (or linear programming) is a mathematical method for determining the value of decision variables in order to obtain the best outcome (e.g the highest profit or lowest cost) under given constraints We will illustrate the method with a transportation problem described in Case Study A.1 Case study A.1 A company has a transportation problem It needs to transfer two products, nails and screws, to a warehouse The company owns a small van, which can transport a maximum of tonnes in weight and 10 m3 in volume of goods The transport needs to take into account the following data: tonne of nails has a volume of m3 and brings in a revenue of £200; tonne of screws has a volume of m3 and brings in a revenue of £300 What quantity of each product should the company transport in order to maximize the benefit? The solution to a linear programming problem is achieved in four steps: Identify the objective of the problem The objective of the problem is to either maximize (e.g profit) or minimize (e.g costs) a function In Case Study A.1, the objective is to maximize the benefit Indentify the decision variables Decision variables are independent variables that are changed until the desired benefit is obtained (i.e maximum or minimum) In Case Study A.1, the weight Multi-Criteria Decision Analysis: Methods and Software, First Edition Alessio Ishizaka and Philippe Nemery © 2013 John Wiley & Sons, Ltd Published 2013 by John Wiley & Sons, Ltd www.it-ebooks.info APPENDIX: LINEAR OPTIMIZATION 289 of the nails and screws should be adjusted to maximize the benefit under the given constraints Identify the objective function The objective function describes a linear relation between the decision variables and the objective of the problem In Case Study A.1, we need to maximize the weight of nails (x) and screws (y) sold in order to maximize the profit: max 200x + 300y Identify the constraints The constraints define the limit of the decision variables They are also linear In Case Study A.1, the constraints are set by the capacity of the van: x+y≤5 x + 5y ≤ 10 x, y ≥ A.2 (constraint on weight) (constraint on volume) (non-negative constraint) Graphical solution If the problem only contains two decision variables, a solution can be found graphically Each axis represents a decision variable and the straight lines of the constraints, obtained by replacing the inequality with equality, are sketched: y = −x + y = (−x + 10)/5 (constraint on weight) (constraint on volume) These lines determine the feasible region, which is the collection of all the points that satisfy all constraints The direction of the arrows (left or right of the lines) is decided by testing one point, generally the origin If this point satisfies the constraint, then the arrow points in that direction For example, if we introduce the origin (0,0) in the constraint weight (0 ≤ 5) and volume (0 ≤ 10), both constraints are satisfied Finally, the coordinates of each corner point should be substituted into the objective function to determine the optimal value because the solution is necessarily on one of the corner point In Case Study A.1, the optimal value is 3.75 tonnes of nails and 1.25 tonnes of screws (Figure A.1) For an analytic solution, where there are more than two decision variables, the simplex algorithm is used A.3 Solution with Microsoft Excel In Figure A.2 the problem is modelled in Microsoft Excel The first three lines are the given data The variable parameters (Figure A.2) have to be entered in Solver (Figure A.3): www.it-ebooks.info 290 APPENDIX: LINEAR OPTIMIZATION x + y ≤ (weight) y (screws) x + 5y ≤ 10 (volume) max z = 200x + 300y optimum (3.75;1.25) x (nails) Figure A.1 Graphical solution r The objective of the problem to be maximized is given by the benefit in cell D12 r The decision variables are set in cells B6 and B7 r The constraint on weight is given by cell B12 (which must be less than or equal to B14) r The constraint on volume is given by cell C12 (which must be less than or equal to C14) The Solver changes the initial data in B6 and B7 until the maximum in D12 is obtained Figure A.2 Modelling of the problem in Microsoft Excel www.it-ebooks.info APPENDIX: LINEAR OPTIMIZATION 291 Figure A.3 Solver parameters Exercise A.1 Here you will learn to use the Microsoft Excel Solver Learning Outcomes Understand the modelling of a linear optimization problem Understand the configuration of Microsoft Excel Solver Tasks Open the file Transport.xls It contains a spreadsheet with the modelling of the problem of the Case Study A.1 Answer the following questions: a) In the spreadsheet, find the objective of the problem, the decision variables and the constraints (Read the comments in the red square in case of difficulty.) b) Open the Solver What is entered in the set target cell? What is entered in the ‘By Changing Cells’ box? What is entered in the ‘Subject to Constraints’ box? www.it-ebooks.info Index 123AHP, 20 additive aggregation, 39 additive model, 84 AHPSort, 53 analytic hierarchy process ordering, 44 anti-ideal solution, 215 assignment, 207 benchmarks, 246 benefit hierarchy, 41 central profile, 51, 52, 173 Chebyshev goal programming, 232 choice problem, 3, 181 ChoiceResults, 20 classification problems, closed system, 39 closeness coefficient, 219 cluster, 59 clusters and pivots method, 51 comparative judgment, 118 comparison matrix, 18 comparisons missing, 45 compensation, 180 complete ranking, 145, 166 concordance, 185 concordance index, 195 Condorcet paradox, 203 consensus vote, 44 consistency, 19, 33, 133 consistency ratio, 23 constant return to scale, 266 correlated criteria, 59 correlation, 146 cost efficiency, 249 cost hierarchy, 41 credibility index, 195 criteria priorities, 16 Criterium, 20 CRS efficient frontier, 239 CRS model, 237 DEAP, 253 DECERNS, 286 Decision Deck, 278 Decision Lens, 20, 68 decision stick, 147, 157, 168 description problem, design problem, deviational variable, 223 direct rating, 21 discordance index, 196 diseconomies of scale, 244, 250 distillation, 187 distillation, 200 distributive mode, 39 distributive normalisation, 217 DMU, 236, 247 Multi-Criteria Decision Analysis: Methods and Software, First Edition Alessio Ishizaka and Philippe Nemery © 2013 John Wiley & Sons, Ltd Published 2013 by John Wiley & Sons, Ltd www.it-ebooks.info 294 INDEX EasyMind, 20 economies of scale, 244, 250 efficiency frontier, 236 efficient frontier, 127 eigenvalue method, 38 ELECTRE I, 181 ELECTRE II, 182 ELECTRE III-IV software, 194 ELECTRE IS, 181 ELECTRE Iv, 181 ELECTRE Tri, 209 elicitation problem, elimination problem, environmental variables, 268 Expert Choice, 20 exponential marginal utility, 86 ideal solution, 215 incoherence, 119 incomparability, 42, 163 incomparable, 145, 157, 200 incompatibility, 120, 125 inconsistency, 125, 133 inconsistent, 23 indifference, 100, 145, 163, 181 indifference relation, 82 indifference threshold, 142, 185 inflexion point, 139 influence matrix, 67, 70 inner dependency, 64, 70 input orientation, 238 interval scale, 114, 181 feedback, 66 flow, 146 FlowSort, 176 FSGaia, 174 fundamental scale, 17 GAHP, 48 GAIA, 147, 157, 168, 172 Gaussian preference function, 139, 163 GDSS, 284 geometric mean, 39 GIS, 279 global alternative priorities, 16 global concordance degree, 196 global flow, 146 global priority, 24 goal, 65, 70 GRIP, 112 group analytic hierarchy process, 48 group decision, 95, 172 lexicographic goal programming, 230 limiting profile, 51, 52, 173, 204 linear optimisation, 291 linear preference function, 162 linear programme, 99, 101, 131 linear programming, 288 local alternative priorities, 16 logarithmic least squares method, 38 Malmquist index, 270 marginal utility, 82 marginal utility functions, 90 Markov chain, 78 missing value, 150 monotonicity, 101 multiplicative aggregation, 40 hard constraint, 223 hierarchy, 14, 27, 90 hierarchy, 15 HIPRE 3+, 20 nadir, 218 negative flow, 143, 164 negative ideal, 218 net flow, 143, 165 node, 122 non-criteria node, 115 normalisation, 85 distributive, 217 ideal, 217 ideal mode, 39 ideal normalisation, 217 objective function, 289 open system, 39 www.it-ebooks.info INDEX optimistic assignment, 207 ordinal regression, 111 outer dependency, 66, 70 output orientation, 238 outranking degree, 187, 195, 197 outranking relation, 183, 194 pairwise, 139 pairwise comparison, 17, 114 Pareto front, 225 partial concordance degree, 195 partial discordance degree, 196 partial ordinal ranking, 42 partial ranking, 166 peers, 246 pessimistic assignment, 207 pie chart, 25 pivot, 48 positive flow, 142, 164 power method, 36 preference, 100 preference degree, 142, 165 preference function, 162 preference relation, 82 preference threshold, 142, 185 preference value functions, 283 pre-order, 187 principal component analysis, 167 priority, 39 problem structuring, 27, 69, 115, 122 profit efficiency, 249 PROMETHEE GDSS, 172 PROMETHEE I, 145, 166 PROMETHEE II, 145, 166 PromSort, 173 pseudo-criteria, 182 Questfox, 20 rank reversal, 37, 39, 169, 203 ranking problem, ratio scale, 28, 114 reciprocity, 31 295 reduced outranking graph, 202 rescaling, 85 returns to scale, 246 revenue efficiency, 249 Right Choice, 97 right–left inconsistency, 37 RightChoiceDSS, 20 robustness analysis, 127 Saaty scale, 19 scale, 31 asymptotic, 30 balanced, 30 geometric, 29 qualitative, 117 root square, 29 scale efficiency, 242 semantic inconsistency, 120, 126 semantic judgment, 118 sensitivity analysis, 19, 25, 74, 94, 127, 148 shop location, 16 slack, 246 Smart Picker Pro, 149–160 soft constraint, 223 solver, 132, 228, 289 sorting, 53, 176, 209 sorting problem, spider diagram, 25 stacked bar diagrams, 24 sub-criteria, 14, 15 SuperDecisions, 76 supermatrix, 80 supra decision-maker, 47 technical efficiency, 248 thermometer view, 156 threshold, 142 time series data, 270 Tobit regression, 268 transitivity, 31, 82, 100 unicriterion preference degree, 139 UTA, 105 www.it-ebooks.info 296 INDEX UTA+, 99 UTAgms, 111 utility function 81 VisualUTA, 108 VRS efficient frontier, 240 VRS model, 237 value tree, 115, 122 variable returns to scale, 267 Ventana Systems, 89 verbal comparison, 28 verbal scale, 152 veto, 182 veto threshold, 197 Visual Promethee, 149 walking weights, 92 weight, 29, 47, 112, 153, 268 weighted goal programming, 228 weighted sum, 2, 21, 72, 83 Win4DEAP, 253 window analysis, 270 zenith, 218 www.it-ebooks.info ... a decision To build long-term relationships, make sustainable and environmentally friendly decisions, companies consider multiple criteria in their decision process Multi-Criteria Decision Analysis: .. .Multi-Criteria Decision Analysis Methods and Software Alessio Ishizaka Reader in Decision Analysis, Portsmouth Business School University of... Published 2013 by John Wiley & Sons, Ltd www.it-ebooks.info MULTI-CRITERIA DECISION ANALYSIS Table 1.1 Category of decision problems Decision Time perspective Novelty Degree of structure Automation